Algebra Examples

$- x^{3} - 4 = 0.5 x^{2} + x + 4$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$0.5 x^{2} + x + 4 = - x^{3} - 4$

Step 2

Add $x^{3}$ to both sides of the equation.

$0.5 x^{2} + x + 4 + x^{3} = - 4$

Step 3

Add $4$ to both sides of the equation.

$0.5 x^{2} + x + 4 + x^{3} + 4 = 0$

Step 4

Add $4$ and $4$ .

$0.5 x^{2} + x + x^{3} + 8 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $x$ out of $0.5 x^{2} + x$ .

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Step 5.1.1

Factor $x$ out of $0.5 x^{2}$ .

$x (0.5 x) + x + x^{3} + 8 = 0$

Step 5.1.2

Raise $x$ to the power of $1$ .

$x (0.5 x) + x + x^{3} + 8 = 0$

Step 5.1.3

Factor $x$ out of $x^{1}$ .

$x (0.5 x) + x \cdot 1 + x^{3} + 8 = 0$

Step 5.1.4

Factor $x$ out of $x (0.5 x) + x \cdot 1$ .

$x (0.5 x + 1) + x^{3} + 8 = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor $0.5$ out of $0.5 x + 1$ .

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Step 5.2.1.1

Factor $0.5$ out of $0.5 x$ .

$x (0.5 (x) + 1) + x^{3} + 8 = 0$

Step 5.2.1.2

Factor $0.5$ out of $1$ .

$x (0.5 x + 0.5 \cdot 2) + x^{3} + 8 = 0$

Step 5.2.1.3

Factor $0.5$ out of $0.5 x + 0.5 \cdot 2$ .

$x (0.5 (x + 2)) + x^{3} + 8 = 0$

Step 5.2.2

Remove unnecessary parentheses.

$x \cdot (0.5 (x + 2)) + x^{3} + 8 = 0$

Step 5.3

Rewrite $8$ as $2^{3}$ .

$x \cdot (0.5 (x + 2)) + x^{3} + 2^{3} = 0$

Step 5.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

$x \cdot (0.5 (x + 2)) + (x + 2) (x^{2} - x \cdot 2 + 2^{2}) = 0$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $2$ by $- 1$ .

$x \cdot (0.5 (x + 2)) + (x + 2) (x^{2} - 2 x + 2^{2}) = 0$

Step 5.5.2

Raise $2$ to the power of $2$ .

$x \cdot (0.5 (x + 2)) + (x + 2) (x^{2} - 2 x + 4) = 0$

Step 5.6

Factor $x + 2$ out of $x \cdot 0.5 (x + 2) + (x + 2) (x^{2} - 2 x + 4)$ .

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Step 5.6.1

Factor $x + 2$ out of $x \cdot 0.5 (x + 2)$ .

$(x + 2) (x \cdot 0.5) + (x + 2) (x^{2} - 2 x + 4) = 0$

Step 5.6.2

Factor $x + 2$ out of $(x + 2) (x \cdot 0.5) + (x + 2) (x^{2} - 2 x + 4)$ .

$(x + 2) (x \cdot 0.5 + x^{2} - 2 x + 4) = 0$

Step 5.7

Move $0.5$ to the left of $x$ .

$(x + 2) (0.5 x + x^{2} - 2 x + 4) = 0$

Step 5.8

Subtract $2 x$ from $0.5 x$ .

$(x + 2) (x^{2} - 1.5 x + 4) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x^{2} - 1.5 x + 4 = 0$

Step 7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

Set $x^{2} - 1.5 x + 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x^{2} - 1.5 x + 4$ equal to $0$ .

$x^{2} - 1.5 x + 4 = 0$

Step 8.2

Solve $x^{2} - 1.5 x + 4 = 0$ for $x$ .

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Step 8.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.2.2

Substitute the values $a = 1$ , $b = - 1.5$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{1.5 \pm \sqrt{{(- 1.5)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Simplify the numerator.

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Step 8.2.3.1.1

Raise $- 1.5$ to the power of $2$ .

$x = \frac{1.5 \pm \sqrt{2.25 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1.5 \pm \sqrt{2.25 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{1.5 \pm \sqrt{2.25 - 16}}{2 \cdot 1}$

Step 8.2.3.1.3

Subtract $16$ from $2.25$ .

$x = \frac{1.5 \pm \sqrt{- 13.75}}{2 \cdot 1}$

Step 8.2.3.1.4

Rewrite $- 13.75$ as $- 1 (13.75)$ .

$x = \frac{1.5 \pm \sqrt{- 1 \cdot 13.75}}{2 \cdot 1}$

Step 8.2.3.1.5

Rewrite $\sqrt{- 1 (13.75)}$ as $\sqrt{- 1} \cdot \sqrt{13.75}$ .

$x = \frac{1.5 \pm \sqrt{- 1} \cdot \sqrt{13.75}}{2 \cdot 1}$

Step 8.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1.5 \pm i \sqrt{13.75}}{2 \cdot 1}$

Step 8.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1.5 \pm i \sqrt{13.75}}{2}$

Step 8.2.4

The final answer is the combination of both solutions.

$x = \frac{1.5 + i \sqrt{13.75}}{2}, \frac{1.5 - i \sqrt{13.75}}{2}$

Step 9

The final solution is all the values that make $(x + 2) (x^{2} - 1.5 x + 4) = 0$ true.

$x = - 2, \frac{1.5 + i \sqrt{13.75}}{2}, \frac{1.5 - i \sqrt{13.75}}{2}$